where \(\text{Diffeo}^+(S_g)\)  is the group of orientation-preserving diffeomorphisms of \(S_g\) , and \(\text{Diffeo}_0^+(S_g)\)  denotes the connected component of the identity.
The mapping class group is of fundamental importance in low-dimensional topology, and has been studied for more than a century. Excellent surveys on recent progress and open problems related to the group \(\Gamma _g\)  can be found in [1]}, [2]}, and a comprehensive treatment is given in [3]}.
Recently, and perhaps surprisingly, ideas from quantum field theory have been applied very successfully to answer questions about the group \(\Gamma _g\) , which
resisted all traditional methods.
